The Modigliani and Miler formulae can be used to re-calculate the cost of equity and the WACC in a company where the level of gearing changes, provided there is no change in the overall business risk and the company is therefore similar in all respects except for its gearing.
When a company plans a new capital investment that will alter its gearing, without affecting its business risk profile, the MM formulae can be used to calculate the cost of equity and WACC at the new level of gearing. The new WACC can then be used as the discount rate for calculating the NPV of the proposed project.
If you are given an examination question of this type in your examination, you might prefer to use the formula for Proposition 2 (cost of equity) because this is given to you in the formula sheet in the examination, and you do not need to learn it.
Study the following example carefully.
Example
A US company has a current total market value of $12 billion, consisting of $10 billion of equity and $2 billion of debt capital. The debt capital will mature in four years’ time. The current weighted average cost of capital is 6.5%.
The company is considering a new investment costing $3 billion, which it would finance entirely by $3 billion of new ten-year bonds.
The yield curve for US government bonds (Treasuries) shows that the risk-free cost of four-year debt is 4% and the risk-free cost of ten-year debt is 4.25%. The credit rating on the company’s current debt capital is AA, but if the new bond issue takes place there is a 75% probability that all the company’s debt will be re-rated to AA- and a 25% probability that all the company’s debt will be re-rated to A+. (This applies to both the existing debt and the new bonds.)
The spreads for yields on corporate bonds above the US Treasuries yield curve are as follows:
Credit rating Spreads (basis points) 4-year bonds 10-year bonds
AA 30 50
AA - 40 60
A + 45 70
The rate of taxation on company profits is 30%
Required
Calculate the weighted average cost of capital in the company if the project goes ahead and is financed entirely by ten-year bonds. Assume that there will be no change in the company’s business risk.
Answer
Total value of the company
In the previous example, it was possible to re-calculate the total value of the company using the Modigliani and Miller formula VG=VU+Dt .
In this example, this is not possible, because the company is raising additional debt capital. The MM formula applies only when the size of the company is unchanged, and equity replaces debt or debt replaces equity. The formula cannot be used when additional debt capital is raised.
An assumption will therefore be made (even though it is unsatisfactory) that when the company obtains its additional debt capital of $3 billion, the value of the equity will be unchanged at $10 billion.
Preliminary workings
We need to calculate the current cost of equity in the company, in order to use the MM formula for Proposition 2. We know the current WACC which is 6.5%. We also know that the current credit rating for the four-year debt capital is AA, which means that the pre-tax cost of debt is the risk-free yield of 4.00% plus a spread of 30 basis points. The pre-tax cost of debt is therefore 4.30% and the after-tax cost of debt is 4.30% (1 – 0.30) = 3.01%.
We now have enough data to calculate the current cost of equity. The WACC, which is 6.5%, is calculated as follows:
Source of finance Market value
Cost Market value
× Cost
$ billion % MV × r
Equity 10.0 r 10r
Debt 2.0 3.01 6.02
12.0 10r + 6.02
WACC = 6.5% = %
12 10r +6.02 6.5 (12) = 10r + 6.02 10r = 71.98
r = 7.198%
The current cost of equity is 7.198%.
We also need to calculate the cost of debt in the company after the new bond issue.
If AA - If A +
4 year 10 year 4 year 10 year
Risk-free 4.00 4.25 4.00 4.25
Spread 0.40 0.60 0.45 0.70
Pre-tax cost 4.40 4.85 4.45 4.95
We can calculate a weighted average pre-tax cost of the debt capital, as follows:
Market value AA - A +
$bn Cost Cost × MV Cost Cost × MV
4 year 2 4.40 8.80 4.45 8.90
10 year 3 4.85 14.55 4.95 14.85
5 23.35 23.75
Weighted average before-tax cost of debt if credit rating is AA - = 23.35/5 = 4.67%.
Weighted average before-tax cost of debt if credit rating is A + = 23.75/5 = 4.75%.
Expected weighted average before tax cost of debt = (0.75 × 4.67) + (0.25 × 4.75)%
= 4.69%.
Expected weighted average after-tax cost of debt = 4.69% (1 – 0.30) = 3.283%.
Calculate the cost of equity in the company after the new bond issue
Step 1. Calculate the cost of equity in an ungeared company at the current level of gearing. We know that the current cost of equity at the current level of gearing is 7.198%.
( )
E K D K ) 1 ( K
KEG = EU + −t EU − D
7.198 = KEU + [(1 – 0.30) (KEU – 4.30) 2/10]
7.198 = KEU + 0.14 KEU – 0.602 1.14 KEU = 7.8
KEU = 6.842.
Step 2. Calculate the cost of equity in the company at the new level of gearing, making the assumption about market value discussed above, but assuming no change in the cost of debt. (Changing the cost of debt in this calculation would produce an incorrect valuation for the cost of equity.)
( )
E K D K ) 1 ( K
KEG = EU + −t EU − D KEG = 6.842 + 0.890
KEG = 7.732.
Calculate the WACC at the new level of gearing We can now calculate the WACC after the bond issue.
Source of finance Market value
Cost Market value
× Cost
$ billion % MV × r
Equity 10.0 7.732 77.320
Debt (weighted average) 5.0 3.283 16.415
15.0 93.735
WACC = 93.735/15 = 6.249, say 6.25%.
A discount rate of 6.25% should be used to calculate the NPV of the proposed new capital investment.
Asset beta
Changes in gearing, the cost of equity and equity beta factors
Definition of asset beta
The asset beta formula
Using the asset beta formula
The asset beta formula: changing from one level of gearing to another
Calculating a project beta factor
Estimating an equity beta factor for a private company
7 Asset beta